Low-noise RF oscillation and optical comb generation based on nonlinear optical resonator

ABSTRACT

Techniques and devices based on optical resonators made of nonlinear optical materials and nonlinear wave mixing to generate RF or microwave oscillations and optical comb signals.

PRIORITY CLAIMS AND RELATED PATENT APPLICATIONS

This patent document claims the priority of U.S. Provisional ApplicationNo. 61/033,409 entitled “LOW-NOISE RF OSCILLATORS BASED ON PHOTONICFEEDBACK LOOP AND NONLINEAR OPTICAL RESONATOR” and filed on Mar. 3,2008, U.S. Provisional Application No. 61/041,071 entitled “TUNABLEFREQUENCY COMB AND MICROWAVE PHOTONIC OSCILLATOR BASED ON FOUR WAVEMIXING IN MEDIA POSSESSING CUBIC NONLINEARITIES” and filed on Mar. 31,2008, and U.S. Provisional Application No. 61/083,852 entitled“INJECTION LOCKING OF HYPERPARAMETRIC OPTICAL COMBS FOR GENERATIONSTABLE MICROWAVE AND RF SIGNALS” and filed on Jul. 25, 2008. The entiredisclosures of the above patent applications are incorporated byreference as part of the disclosure of this document.

BACKGROUND

This application relates to signal oscillators based on photonicdevices.

RF and microwave oscillators for generating signals in the RF andmicrowave frequencies may be constructed as “hybrid” devices by usingboth electronic and optical components to form opto-electronicoscillators (“OEOs”). See, e.g., U.S. Pat. Nos. 5,723,856, 5,777,778,5,929,430, and 6,567,436. Such an OEO includes an electricallycontrollable optical modulator and at least one active opto-electronicfeedback loop that comprises an optical part and an electrical partinterconnected by a photodetector. The opto-electronic feedback loopreceives the modulated optical output from the modulator and convertedthe modulated optical output into an electrical signal which is appliedto control the modulator. The feedback loop produces a desired longdelay in the optical part of the loop to suppress phase noise and feedsthe converted electrical signal in phase to the modulator to generatethe optical modulation and generate and sustain an electricaloscillation in RF or microwave frequencies when the total loop gain ofthe active opto-electronic loop and any other additional feedback loopsexceeds the total loss. The generated oscillating signals are tunable infrequency and can have narrow spectral linewidths and low phase noise incomparison with the signals produced by other RF and microwavesoscillators.

SUMMARY

This document provides techniques and devices based on opticalresonators made of nonlinear optical materials and nonlinear wave mixingto generate RF or microwave oscillations and optical comb signals.

In one aspect, this document provides a method for generating an RF ormicrowave oscillation. This method includes coupling a laser beam from alaser into a whispering-gallery-mode resonator formed of a nonlinearoptical material with a third order nonlinearity at a power level abovea pump threshold power level to cause an optical hyperparametricoscillation based on a nonlinear mixing in the resonator; and couplinglight out of the resonator into a photodetector to produce an RF ormicrowave signal at an RF or microwave frequency with low noise.

In another aspect, a device is provided for generating an RF ormicrowave oscillation. This device includes a whispering-gallery-moderesonator formed of a nonlinear optical material exhibiting a thirdorder nonlinearity; a laser to produce a laser beam that interacts withthe nonlinear optical material to cause an optical hyperparametricoscillation due to a nonlinear mixing in the resonator; a first opticalcoupler that directly couples to the laser at one end of the firstoptical coupler to receive the laser beam and directly couples to theresonator to couple the laser beam into the resonator; a second opticalcoupler that directly couples to the resonator to output light out ofthe resonator; and a photodetector directly coupled to the secondoptical coupler to receive the output light and to produce an opticalhyperparametric oscillation signal at an RF or microwave frequency withlow noise.

In another aspect, a method is provided for generating an optical combsignal carrying different optical frequencies. This method includescoupling an optical pump beam from a laser into awhispering-gallery-mode resonator formed of a nonlinear optical materialwith a third order nonlinearity at a power level above a pump thresholdpower level to cause an optical hyperparametric oscillation based on anonlinear mixing in the resonator to produce optical sidebands; andcoupling light out of the resonator to generate an optical comb signalhaving the optical sidebands and a band at a frequency of the opticalpump.

In yet another aspect, a device is provided to include an opticalresonator formed of a nonlinear optical material exhibiting a thirdorder nonlinearity and structured to circulate light in one or moreresonator modes; a laser to produce a laser beam that interacts with thenonlinear optical material to cause an optical hyperparametricoscillation due to a nonlinear mixing in the resonator; an opticalcoupler that couples to the resonator to couple the laser beam from thelaser along a first optical path into the resonator and couples lightinside the resonator out as a resonator output beam along a secondoptical path; an optical reflector in the second optical path to reflectat least a portion of the resonator output beam back to the opticalcoupler which couples the reflected portion into the resonator in a waythat the optical coupler couples light of the reflected portion insidethe resonator out as a feedback beam to the laser; and a mechanism tocontrol a phase of the feedback beam to establish a phase matching forlocking the laser to the resonator based on injection locking.

These and other aspects and implementations are described in detail inthe drawings, the description and the claims.

BRIEF DESCRIPTION OF DRAWING

FIGS. 1, 2, 3, 4A, 4B, 5A and 5B show examples of WGM resonators andoptical coupling designs.

FIGS. 6A, 6B and 6C show examples of RF or microwave oscillators basedon nonlinear WGM resonators.

FIGS. 7 and 8 illustrate operations of nonlinear WGM resonators underoptical pumping.

FIG. 9 shows a Pound-Drever-Hall (PDH) laser feedback locking scheme forlocking a laser to a nonlinear WGM resonator.

FIGS. 10-15 show measurements of sample nonlinear WGM resonators forgenerating optical comb signals.

FIG. 16 shows an example for locking a laser to a resonator by using anexternal reflector.

DETAILED DESCRIPTION

This application describes implementations of a high frequency photonicmicrowave oscillator (e.g., in the X-W bands) based on the nonlinearprocess of four wave mixing (FWM) in crystalline whispering gallery moderesonators such as calcium fluoride or another material possessing cubicnonlinearity) that can be packaged in small packages. In FWM, the largefield intensity in the high finesse WGM transforms two pump photons intotwo sideband photons, i.e., a signal photo and an idler photon. The sumof frequencies of the generated photons is equal to twice the frequencyof the pumping light because of the energy conservation law. Bysupersaturating the oscillator and using multiple optical harmonicsescaping the resonator (optical comb) the described oscillators canreduce the phase noise and increase spectral purity of the microwavesignals generated on a fast photodiode.

The optical resonators may be configured as opticalwhispering-gallery-mode (“WGM”) resonators which support a special setof resonator modes known as whispering gallery (“WG”) modes. These WGmodes represent optical fields confined in an interior region close tothe surface of the resonator due to the total internal reflection at theboundary. For example, a dielectric sphere may be used to form a WGMresonator where WGM modes represent optical fields confined in aninterior region close to the surface of the sphere around its equatordue to the total internal reflection at the sphere boundary. Quartzmicrospheres with diameters on the order of 10˜10² microns have beenused to form compact optical resonators with Q values greater than 10⁹.Such hi-Q WGM resonators may be used to produce oscillation signals withhigh spectral purity and low noise. Optical energy, once coupled into awhispering gallery mode, can circulate at or near the sphere equatorover a long photon life time.

WGM resonators made of crystals described in this application can beoptically superior to WGM resonators made of fused silica. WGMresonators made of crystalline CaF₂ can produce a Q factor at or greaterthan 10¹⁰. Such a high Q value allows for various applications,including generation of kilohertz optical resonances and low-thresholdoptical hyperparametric oscillations due to the Kerr nonlinear effect.The following sections first describe the exemplary geometries forcrystal WGM resonators and then describe the properties of WGMresonators made of different materials.

FIGS. 1, 2, and 3 illustrate three exemplary WGM resonators. FIG. 1shows a spherical WGM resonator 100 which is a solid dielectric sphere.The sphere 100 has an equator in the plane 102 which is symmetric aroundthe z axis 101. The circumference of the plane 102 is a circle and theplane 102 is a circular cross section. A WG mode exists around theequator within the spherical exterior surface and circulates within theresonator 100. The spherical curvature of the exterior surface aroundthe equator plane 102 provides spatial confinement along both the zdirection and its perpendicular direction to support the WG modes. Theeccentricity of the sphere 100 generally is low.

FIG. 2 shows an exemplary spheroidal microresonator 200. This resonator200 may be formed by revolving an ellipse (with axial lengths a and b)around the symmetric axis along the short elliptical axis 101 (z).Therefore, similar to the spherical resonator in FIG. 1, the plane 102in FIG. 2 also has a circular circumference and is a circular crosssection. Different from the design in FIG. 1, the plane 102 in FIG. 2 isa circular cross section of the non-spherical spheroid and around theshort ellipsoid axis of the spheroid. The eccentricity of resonator 100is (1−b²/a²)^(1/2) and is generally high, e.g., greater than 10⁻¹.Hence, the exterior surface is the resonator 200 is not part of a sphereand provides more spatial confinement on the modes along the z directionthan a spherical exterior. More specifically, the geometry of the cavityin the plane in which Z lies such as the zy or zx plane is elliptical.The equator plane 102 at the center of the resonator 200 isperpendicular to the axis 101 (z) and the WG modes circulate near thecircumference of the plane 102 within the resonator 200.

FIG. 3 shows another exemplary WGM resonator 300 which has anon-spherical exterior where the exterior profile is a general conicshape which can be mathematically represented by a quadratic equation ofthe Cartesian coordinates. Similar to the geometries in FIGS. 1 and 2,the exterior surface provides curvatures in both the direction in theplane 102 and the direction of z perpendicular to the plane 102 toconfine and support the WG modes. Such a non-spherical, non-ellipticalsurface may be, among others, a parabola or hyperbola. Note that theplane 102 in FIG. 3 is a circular cross section and a WG mode circulatesaround the circle in the equator.

The above three exemplary geometries in FIGS. 1, 2, and 3 share a commongeometrical feature that they are all axially or cylindrically symmetricaround the axis 101 (z) around which the WG modes circulate in the plane102. The curved exterior surface is smooth around the plane 102 andprovides two-dimensional confinement around the plane 102 to support theWG modes.

Notably, the spatial extent of the WG modes in each resonator along thez direction 101 is limited above and below the plane 102 and hence itmay not be necessary to have the entirety of the sphere 100, thespheroid 200, or the conical shape 300. Instead, only a portion of theentire shape around the plane 102 that is sufficiently large to supportthe whispering gallery modes may be used to form the WGM resonator. Forexample, rings, disks and other geometries formed from a proper sectionof a sphere may be used as a spherical WGM resonator.

FIGS. 4A and 4B show a disk-shaped WGM resonator 400 and a ring-shapedWGM resonator 420, respectively. In FIG. 4A, the solid disk 400 has atop surface 401A above the center plane 102 and a bottom surface 401Bbelow the plane 102 with a distance H. The value of the distance H issufficiently large to support the WG modes. Beyond this sufficientdistance above the center plane 102, the resonator may have sharp edgesas illustrated in FIGS. 3, 4A, and 4B. The exterior curved surface 402can be selected from any of the shapes shown in FIGS. 1, 2, and 3 toachieve desired WG modes and spectral properties. The ring resonator 420in FIG. 4B may be formed by removing a center portion 410 from the soliddisk 400 in FIG. 4A. Since the WG modes are present near the exteriorpart of the ring 420 near the exterior surface 402, the thickness h ofthe ring may be set to be sufficiently large to support the WG modes.

An optical coupler is generally used to couple optical energy into orout of the WGM resonator by evanescent coupling. FIGS. 5A and 5B showtwo exemplary optical couplers engaged to a WGM resonator. The opticalcoupler may be in direct contact with or separated by a gap from theexterior surface of the resonator to effectuate the desired criticalcoupling. FIG. 5A shows an angle-polished fiber tip as a coupler for theWGM resonator. A waveguide with an angled end facet, such as a planarwaveguide or other waveguide, may also be used as the coupler. FIG. 5Bshows a micro prism as a coupler for the WGM resonator. Other evanescentcouplers may also be used, such as a coupler formed from a photonicbandgap material.

WGM resonators can be used to provide an effective way to confinephotons in small volumes for long periods of time. As such, WGMresonators have a wide range of applications in both fundamental studiesand practical devices. For example, WGM resonators can be used forstorage of light with linear optics, as an alternative to atomic lightstorage, as well as in tunable optical delay lines, a substitute foratomic-based slow light experiments. WGM resonators can also be used foroptical filtering and opto-electronic oscillators, among otherapplications.

Amongst many parameters that characterize a WGM resonator (such asefficiency of in and out coupling, mode volume, free spectral range,etc.) the quality factor (Q) is a basic one. The Q factor is related tothe lifetime of light energy in the resonator mode (τ) as Q=2πυτ, wherev is the linear frequency of the mode. The ring down time correspondingto a mode with Q=2×10¹⁰ and wavelength λ=1.3 μm is 15 μs, thus makingultrahigh Q resonators potentially attractive as light storage devices.Furthermore, some crystals are transparent enough to allow extremelyhigh-Q whispering gallery modes while having important nonlinearproperties to allow continuous manipulation of the WGMs' characteristicsand further extend their usefulness.

In a dielectric resonator, the maximum quality factor cannot exceedQ_(max)=2πn₀/(λα), where no is the refractive index of the material, λis the wavelength of the light in vacuum, and α is the absorptioncoefficient of the dielectric material. The smaller the absorption, thelarger is Q_(max). Hence, to predict the narrowest possible linewidthγ=τ⁻¹ of a WGM one has to know the value of optical attenuation intransparent dielectrics—within their transparency window—within whichthe losses are considered negligible for the vast majority ofapplications. This question about the residual fundamental absorptionhas remained unanswered for most materials because of a lack ofmeasurement methods with adequate sensitivity. Fortunately, high-Qwhispering gallery modes themselves represent a unique tool to measurevery small optical attenuations in a variety of transparent materials.

Previous experiments with WGM resonators fabricated by thermal reflowmethods applicable to amorphous materials resulted in Q factors lessthan 9×10⁹. The measurements were performed with fused silicamicrocavities, where surface-tension forces produced nearly perfectresonator surfaces, yielding a measured Q factor that approached thefundamental limit determined by the material absorption. It is expectedthat optical crystals would have less loss than fused silica becausecrystals theoretically have a perfect lattice without inclusions andinhomogeneities that are always present in amorphous materials. Thewindow of transparency for many crystalline materials is much wider thanthat of fused silica. Therefore, with sufficiently high-purity material,much smaller attenuation in the middle of the transparency window can beexpected—as both the Rayleigh scattering edge and multiphonon absorptionedge are pushed further apart towards ultraviolet and infrared regions,respectively. Moreover, crystals may suffer less, or not at all, theextrinsic absorption effects caused by chemosorption of OH ions andwater, a reported limiting factor for the Q of fused silica near thebottom of its transparency window at 1.55 μm.

Until recently, one remaining problem with the realization ofcrystalline WGM resonators was the absence of a fabrication process thatwould yield nanometer-scale smoothness of spheroidal surfaces forelimination of surface scattering. Very recently this problem wassolved. Mechanical optical polishing techniques have been used forfabricating ultrahigh-Q crystalline WGM resonators with Q approaching10⁹. In this document, high quality factors (Q=2×10¹⁰) in WGM resonatorsfabricated with transparent crystals are further described.

Crystalline WGM resonators with kilohertz-range resonance bandwidths atthe room temperature and high resonance contrast (50% and more) arepromising for integration into high performance optical networks.Because of small modal volumes and extremely narrow single-photonresonances, a variety of low-threshold nonlinear effects can be observedin WGM resonators based on small broadband nonlinear susceptibilities.As an example, below we report the observation of thermo-opticalinstability in crystalline resonators, reported earlier for much smallervolume high-Q silica microspheres.

There is little consistent experimental data on small opticalattenuation within transparency windows of optical crystals. Forexample, the high sensitivity measurement of the minimum absorption ofspecially prepared fused silica, α=0.2 dB/km at λ=1.55 μm, (Δα≧10⁻⁷cm⁻¹) becomes possible only because of kilometers of optical fibersfabricated from the material. Unfortunately, this method is notapplicable to crystalline materials. Fibers have also been grown out ofcrystals such as sapphire, but attenuation in those (few dB per meter)was determined by scattering of their surface. Calorimetry methods formeasurement of light absorption in transparent dielectrics give an erroron the order of Δα≧10⁻⁷ cm⁻¹. Several transparent materials have beentested for their residual absorption with calorimetric methods, whileothers have been characterized by direct scattering experiments, bothyielding values at the level of a few ppm/cm of linear attenuation,which corresponds to the Q limitation at the level of 10¹⁰. The questionis if this is a fundamental limit or the measurement results werelimited by the imperfection of crystals used.

Selection of material for highest-Q WGM resonators must be based onfundamental factors, such as the widest transparency window, high-puritygrade, and environmental stability. Alkali halides may not be suitabledue to their hygroscopic property and sensitivity to atmospherichumidity. Bulk losses in solid transparent materials can be approximatedwith the phenomenological dependenceα≅α_(UV) e ^(λ) ^(UV) ^(/λ)+α_(R)λ⁻⁴+α_(IR) e ^(−λ) ^(IR) ^(/λ),  (1)where α_(UV), α_(R), and α_(IR) represent the blue wing (primaryelectronic), Rayleigh, and red wing (multiphonon) losses of the light,respectively; λ_(UV), and λ_(IR) stand for the edges of the materialtransparency window. This expression does not take into account resonantabsorption due to possible crystal impurities. Unfortunately,coefficients in Eq. (1) are not always known.

One example of nonlinear materials for fabrication of high-Q WGMresonators with optical nonlinear behaviors is calcium fluoride (CaF₂).This material is useful in various applications because of its use inultraviolet lithography applications at 193 and 157 nm. Ultrapurecrystals of this material suitable for wide aperture optics have beengrown, and are commercially available. According to recently reportedmeasurements on scattering in CaF₂ α=3×10⁻⁵ cm⁻¹ at 193 nm, extremelysmall scattering can be projected in the near-infrared bandcorresponding to the limitation of Q at the level of 10¹³.

Lattice absorption at this wavelength can be predicted from the positionof the middle infrared multiphonon edge, and yields even smaller Qlimitations. Because of residual doping and nonstoichiometry, bothscattering and absorption are present and reduce the Q in actualresonators. An additional source for Q limitation may be the scatteringproduced by the residual surface inhomogeneities resulting from thepolishing techniques. At the limit of conventional optical polishingquality (average roughness σ=2 nm), the estimates based on the waveguidemodel for WGM surface scattering yield Q≅10¹¹.

We studied WGM resonators fabricated with calcium fluoride and a fewother crystalline materials made of LiNbO₃, LiTaO₃ and Al₂O₃, andmeasured their quality factors. CaF₂ resonators were fabricated bycore-drilling of cylindrical preforms and subsequent polishing of therim of the preforms into spheroidal geometry. The fabricated resonatorshad a diameter of 4-7 millimeters and a thickness of 0.5-1 mm. Thefabricated Calcium fluoride resonators had a Q factor of about 2×10¹⁰.

Measurement of the Q was done using the prism coupling method. Theintrinsic Q was measured from the bandwidth of the observed resonancesin the undercoupled regime. Because of different refraction indices inresonators, we used BK7 glass prisms (n=1.52) for silica (n=1.44) andcalcium fluoride (n=1.43), diamond (n=2.36) for lithium niobate (n=2.10,2.20), and lithium niobate prism (n=2.10) for sapphire (n=1.75). We usedextended cavity diode lasers at 760 nm, distributed feedbacksemiconductor lasers at 1550 nm, and solid-state YAG lasers at 1319 nmas the light source.

A high-Q nonlinear WGM resonators can be used for achievinglow-threshold optical hyperparametric oscillations. The oscillationsresult from the resonantly enhanced four-wave mixing occurring due tothe Kerr nonlinearity of the material. Because of the narrow bandwidthof the resonator modes as well as the high efficiency of the resonantfrequency conversion, the oscillations produce stable narrow-bandbeat-note of the pump, signal, and idler waves. A theoretical model forthis process is described.

Realization of efficient nonlinear optical interactions at low lightlevels has been one of the main goals of non-linear optics since itsinception. Optical resonators contribute significantly to achieving thisgoal, because confining light in a small volume for a long period oftime leads to increased nonlinear optical interactions. Opticalwhispering gallery mode (WGM) resonators are particularly well suitedfor the purpose. Features of high quality factors (Q) and small modevolumes have already led to the observation of low-threshold lasing aswell as efficient nonlinear wave mixing in WGM resonators made ofamorphous materials.

Optical hyperparametric oscillations, dubbed as modulation instabilityin fiber optics, usually are hindered by small nonlinearity of thematerials, so high-power light pulses are required for theirobservation. Though the nonlinearity of CaF₂ is even smaller than thatof fused silica, we were able to observe with low-power continuous wavepump light a strong nonlinear interaction among resonator modesresulting from the high Q (Q>5×10⁹) of the resonator. New fields aregenerated due to this interaction.

The frequency of the microwave signal produced by mixing the pump andthe generated side-bands on a fast photodiode is stable and does notexperience a frequency shift that could occur due to the self- andcross-phase modulation effects. Conversely in, e.g., coherent atomicmedia, the oscillations frequency shifts to compensate for the frequencymismatch due to the cross-phase modulation effect (ac Stark shift). Inour system the oscillation frequency is given by the mode structure and,therefore, can be tuned by changing the resonator dimensions. Differentfrom resonators fabricated with amorphous materials and liquids, high-Qcrystalline resonators allow for a better discrimination of thethird-order nonlinear processes and the observation of purehyperparametric oscillation signals. As a result, the hyperoscillator ispromising for applications as an all-optical secondary frequencyreference.

The hyperparametric oscillations could be masked with stimulated Ramanscattering (SRS) and other non-linear effects. For example, anobservation of secondary lines in the vicinity of the optical pumpingline in the SRS experiments with WGM silica microresonators wasinterpreted as four-wave mixing between the pump and two Raman wavesgenerated in the resonator, rather than as the four-photon parametricprocess based on electronic Kerr nonlinearity of the medium. Aninterplay among various stimulated nonlinear processes has also beenobserved and studied in droplet spherical microcavities.

The polarization selection rules together with WGM's geometricalselection rules allow for the observation of nonlinear processesoccurring solely due to the electronic nonlinearity of the crystals incrystalline WGM resonators. Let us consider a fluorite WGM resonatorpossessing cylindrical symmetry with symmetry axis. The linear index ofrefraction in a cubic crystal is uniform and isotropic, therefore theusual description of the modes is valid for the resonator. The TE and TMfamilies of WGMs have polarization directions parallel and orthogonal tothe symmetry axis, respectively. If an optical pumping light is sentinto a TE mode, the Raman signal cannot be generated in the same modefamily because in a cubic crystal such as CaF₂ there is only one, triplydegenerate, Raman-active vibration with symmetry F_(2g). Finally, in theultrahigh Q crystalline resonators, due to the material as well asgeometrical dispersion, the value of the free spectral range (FSR) atthe Raman detuning frequency differs from the FSR at the carrierfrequency by an amount exceeding the mode spectral width. Hence,frequency mixing between the Raman signal and the carrier is stronglysuppressed. Any field generation in the TE mode family is due to theelectronic nonlinearity only, and Raman scattering occurs in the TMmodes.

Consider three cavity modes: one nearly resonant with the pump laser andthe other two nearly resonant with the generated optical sidebands. Ouranalysis begins with the following equations for the slow amplitudes ofthe intracavity fields{dot over (A)}=−Γ ₀ A+ig[|A| ²+2|B ₊|²+2|B ⁻|² ]A+2igA*B ₊ B ⁻ +F ₀,{dot over (B)} ₊=−Γ₊ B ₊ +ig[2|A| ² +|B ₊|²+2|B ⁻|² ]B ₊ +igB* ⁻ |A| ²,{dot over (B)} ⁻=−Γ⁻ B ⁻ +ig[2|A| ²+2|B ₊|² +|B ⁻|² ]B ⁻ +igB* ₊ |A| ²,where Γ_(o)=i(ω_(o)−ω)+γ_(o) and Γ_(±)=i(ω_(±)−{tilde over(ω)}_(±))+γ_(±), γ_(o), γ₊, and γ⁻ as well as ω_(o), ω₊, and ω⁻ are thedecay rates and eigenfrequencies of the optical cavity modesrespectively; ω is the carrier frequency of the external pump (A),{tilde over (ω)}_(±) and {tilde over (ω)}_(±) are the carrierfrequencies of generated light (B₊ and B⁻, respectively). Thesefrequencies are determined by the oscillation process and cannot becontrolled from the outside. However, there is a relation between them(energy conservation law): 2ω={tilde over (ω)}_(±)+{tilde over (ω)}⁻.Dimensionless slowly varying amplitudes A, B₊, and B⁻ are normalizedsuch that |A|², |B₊|², and |B⁻|² describe photon number in thecorresponding modes. The coupling constant can be found from thefollowing expressiong=hω ₀ ² n ₂ c/Vn ₀ ²where n₂ is an optical constant that characterizes the strength of theoptical nonlinearity, n_(o) is the linear refractive index of thematerial, V is the mode volume, and c is the speed of light in thevacuum. Deriving this coupling constant we assume that the modes arenearly overlapped geometrically, which is true if the frequencydifference between them is small. The force F_(o) stands for theexternal pumping of the system F_(o)=(2γ_(o)P_(o)/ω_(o))^(1/2), whereP_(o) is the pump power of the mode applied from the outside.

For the sake of simplicity we assume that the modes are identical, i.e.,y₊=y⁻=y_(o), which is justified by observation with actual resonators.Then, solving the set (1)-(3) in steady state we find the oscillationfrequency for generated fields

${{\omega - {\overset{\sim}{\omega}\mspace{14mu}\ldots}}\mspace{14mu} = {{{\overset{\sim}{\omega}}_{+} - \omega} = {\frac{1}{2}\left( {\omega_{+} - {\omega\mspace{14mu}\ldots}}\mspace{14mu} \right)}}},$i.e., the beat-note frequency depends solely on the frequency differencebetween the resonator modes and does not depend on the light power orthe laser detuning from the pumping mode. As a consequence, theelectronic frequency lock circuit changes the carrier frequency of thepump laser but does not change the frequency of the beat note of thepumping laser and the generated sidebands.

The threshold optical power can be found from the steady state solutionof the set of three equations for the slow amplitudes of the intracavityfields:

${P_{th} \simeq {1.54\mspace{14mu}\frac{\pi}{2}\mspace{14mu}\frac{n_{0}^{2}V}{n_{2}\lambda\; Q^{2}}}},$where the numerical factor 1.54 comes from the influence of theself-phase modulation effects on the oscillation threshold. Thetheoretical value for threshold in our experiment is P_(th)≈0.3 mW,where n_(o)=1.44 is the refractive index of the material, n₂=3.2×10⁻¹⁶cm²/W is the nonlinearity coefficient for calcium fluoride, V=10⁻⁴ cm³is the mode volume, Q=6×10⁹, and λ=1.32 μm.

The above equation suggests that the efficiency of the parametricprocess increases with a decrease of the mode volume. We used arelatively large WGM resonator because of the fabrication convenience.Reducing the size of the resonator could result in a dramatic reductionof the threshold for the oscillation. Since the mode volume may beroughly estimated as V=2πλR², it is clear that reducing the radius R byan order of magnitude would result in 2 orders of magnitude reduction inthe threshold of the parametric process. This could place WGM resonatorsin the same class as the oscillators based on atomic coherence. However,unlike the frequency difference between sidebands in the atomicoscillator, the frequency of the WGM oscillator could be free from power(ac Stark) shifts.

Analysis based on the Langevin equations describing quantum behavior ofthe system suggests that the phase diffusion of the beat-note is small,similar to the low phase diffusion of the hyperparametric process inatomic coherent media. Close to the oscillation threshold the phasediffusion coefficient is

${D_{beat} \simeq {\frac{\gamma_{0}^{2}}{4}\mspace{14mu}\frac{{\hslash\omega}_{0}}{P_{B_{out}}}}},$where P_(Bout) is the output power in a sideband. The correspondingAllan deviation is σ_(beat)/ω_(beat)=(2D_(beat)/tω² _(beat))^(1/2). Wecould estimate the Allan deviation as follows:σ_(beat)/ω_(beat)≅10⁻¹³ /√{square root over (r)}for γ₀=3×10⁵ rad/s, P_(Bout)=1 mW, ω₀=1.4×10¹⁵ rad/s and ω_(beat)=5×10¹⁰rad/s. Follow up studies of the stability of the oscillations in thegeneral case will be published elsewhere.

Our experiments show that a larger number of modes beyond the abovethree interacting modes could participate in the process. The number ofparticipating modes is determined by the variation of the mode spacingin the resonator. Generally, modes of a resonator are not equidistantbecause of the second order dispersion of the material and thegeometrical dispersion. We introduce D=(2ω_(o)−ω₊−ω⁻)/γ_(o) to take thesecond order dispersion of the resonator into account. If |D|≧1 themodes are not equidistant and, therefore, multiple harmonic generationis impossible.

Geometrical dispersion for the main mode sequence of a WGM resonator isD≅0.41c/(γ₀Rn₀m^(5/3)), for a resonator with radius R; ω₊, ω₀, and ω⁻are assumed to be m+1, m, and m−1 modes of the resonator(ω_(m)Rn_(ωm)=mc, m>>1). For R=0.4 cm, γ₀=2×10⁵ rad/s, m=3×10⁴ we obtainD=7×10⁻⁴, therefore the geometrical dispersion is relatively small inour case. However, the dispersion of the material is large enough. Usingthe Sellmeier dispersion equation, we find D≅0.1 at the pump laserwavelength. This implies that approximately three sideband pairs can begenerated in the system (we see only two in the experiment).

Furthermore, the absence of the Raman signal in our experiments showsthat effective Raman nonlinearity of the medium is lower than the valuemeasured earlier. Theoretical estimates based on numbers from predictnearly equal pump power threshold values for both the Raman and thehyperparametric processes. Using the expression derived for SRSthreshold P_(R)≅π2n₀ ²V/Gλ²Q², where G≅2×10⁻¹¹ cm/W is the Raman gaincoefficient for CaF₂, we estimate P_(th)/P_(R)≈1 for any resonator madeof CaF₂. However, as mentioned above, we did not observe any SRS signalin the experiment.

Therefore, because of the long interaction times of the pumping lightwith the material, even the small cubic nonlinearity of CaF₂ results inan efficient generation of narrow-band optical sidebands. This processcan be used for the demonstration of a new kind of an all-opticalfrequency reference. Moreover, the oscillations are promising as asource of squeezed light because the sideband photon pairs generated inthe hyperparametric processes are generally quantum correlated.

Photonic microwave oscillators can be built based on generation andsubsequent demodulation of polychromatic light to produce a well definedand stable beat-note signal. Hyperparametric oscillators based onnonlinear WGM optical resonators can be used to generate ultrastablemicrowave signals. Such microwave oscillators have the advantage of asmall size and low input power, and can generate microwave signals atany desired frequency, which is determined by the size of the resonator.

Hyperparametric optical oscillation is based on four-wave mixing amongtwo pump, signal, and idler photons by transforming two pump photos in apump beam into one signal photon and one idler photon. This mixingresults in the growth of the signal and idler optical sidebands fromvacuum fluctuations at the expense of the pumping wave. A highintracavity intensity in high finesse WGMs results in χ(3)basedfour-photon processes like hω+hω→h(ω+ωM)+h(ω−ωM), where ω is the carrierfrequency of the external pumping, and ωM is determined by the freespectral range of the resonator ωM=ΩFSR. Cascading of the process andgenerating multiple equidistant signal and idler harmonics (opticalcomb) is also possible in this oscillator. Demodulation of the opticaloutput of the oscillator by means of a fast photodiode results in thegeneration of high frequency microwave signals at frequency ωM. Thespectral purity of the signal increases with increasing Q factor of theWGMs and the optical power of the generated signal and idler. Thepumping threshold of the oscillation can be as small as microWatt levelsfor the resonators with ultrahigh Q-factors.

There are several problems hindering the direct applications of thehyperparametric oscillations. One of those problems is related to thefact that the optical signal escaping WGM resonator is mostly phasemodulated. Therefore, a direct detection of the signal on the fastphotodiode does not result in generation of a microwave. To go aroundthis discrepancy, the nonlinear WGM resonator can be placed in an arm ofa Mach-Zehnder interferometer with an additional delay line in anotherarm of the interferometer. The optical interference of the light fromthe two arms allows transforming phase modulated signal into anamplitude modulated signal which can be detected by an optical detectorto produce a microwave signal.

FIG. 6A shows an example of a hyperparametric microwave photonicoscillator in an optical interferometer configuration with a firstoptical path 1611 having the nonlinear WGM resonator 630 and a secondoptical path 612 with a long delay line. Light from a laser 601 is splitinto the two paths 611 and 612. Two coupling prisms 631 and 632 or otheroptical couplers can be used to optically couple the resonator 630 tothe first optical path 611. The output light of the resonator 630 iscollected into a single-mode fiber after the coupling prism 632 and iscombined with the light from the optical delay line. The combined lightis sent to a photodiode PD 650 which produces a beat signal as anarrow-band microwave signal with low noise. A signal amplifier 660 anda spectrum analyzer 660 can be used downstream from the photodiode 650.

FIG. 6B shows an example of a hyperparametric microwave photonicoscillator in which the oscillator is able to generate microwave signalswithout a delay in the above interferometer configuration in FIG. 6A.This simplifies packaging the device.

FIG. 6C shows an oscillator where a laser diode 601 is directly coupledto an optical coupling element CP1 (631, e.g., a coupling prism) that isoptically coupled to the WDM nonlinear resonator 630 and a secondoptical coupling element CP2 (632, e.g., a coupling prism) is coupled tothe resonator 630 to produce an optical output. The photodiode PD 650 iscoupled to the CP2 to convert the optical output received by thephotodiode 650 into a low noise RF/microwave signal.

The above designs without the optical delay line are based on singlesideband four wave mixing process occurring in the resonators. A singlesideband signal does not require any interferometric technique togenerate a microwave signal on the photodiode.

An example of the single-sideband signal is shown in FIG. 7, which showsexperimentally observed spectrum of the hyperparametric oscillator. Theoscillator has only one sideband separated with the carrier by resonatorFSR (12 GHz), unlike to the usual hyperparametric oscillator havingsymmetric sidebands. The optical signal generates 12 GHz spectrally puremicrowave signal on a fast photodiode.

The single-sideband oscillator is suitable for packaging of the devicein a small package. The process occurs due to the presence of multiplefrequencies degenerate optical modes in a WGM resonator. The modesinterfere on the resonator surface. The interference results in specificspatial patterns on the resonator surface. Each sideband generated inthe resonator has its own unique pattern. Selecting the rightgeometrical position of the output coupler on the surface of theresonator, it is possible to retrieve the carrier and only one generatedsideband.

FIG. 8 illustrates that monochromatic light interacting simultaneouslywith several degenerate or nearly degenerate modes of a WGM resonatorresults in the interference pattern on the resonator surface. Thepattern is stationary in time if the modes are completely degenerate.Selecting right position for the output coupler allows detection of theoutput light (e.g., at point A). At the point B, however, there is anull in the optical field so that no light is detected when the coupleris placed at B.

Therefore, a single sideband oscillator can be made by using a nonlinearWGM resonator with comparably high spectral density and an outputevanescent field coupler that can be positioned in the proximity of theresonator surface. We have shown experimentally that by selecting theproper point on the resonator surface it is possible to observe opticalhyperparametric oscillations with only one sideband generated. Such anoscillation can be demodulated directly on a fast photodiode.

The hyperparametric oscillator produces a high spectral purity for themicrowave signal generated at the output of the photodetector. We havemeasured phase noise of the signals and found that it is shot noiselimited and that the phase noise floor can reach at least −126 dBc/Hzlevel. To improve the spectral purity we can oversaturate the oscillatorand generate an optical comb. Microwave signals generated bydemodulation of the optical comb have better spectral purity comparedwith the single-sideband oscillator. Optical comb corresponds to modelocking in the system resulting in generation of short optical pulses.We have found that the phase noise of the microwave signal generated bythe demodulation of the train of optical pulses with duration t andrepetition rate T is given by shot noise with a power spectral densitygiven by

${S_{\phi}(\omega)} \approx {\frac{2{\hslash\omega}_{0}}{P_{ave}\omega^{2}}\mspace{14mu}\frac{4\pi^{2}\alpha\; t^{2}}{T^{4}}}$where ω0 is the frequency of the optical pump, P_(ave) is the averagedoptical power of the generated pulse train, α is the round trip opticalloss. Hence, the shorter is the pulse compared with the repetition ratethe smaller is the phase noise. On the other hand we know that T/t isapproximately the number of modes in the comb N. Hence, we expect thatthe comb will have much lower (N^2) phase noise compared with usualhyperparametric oscillator having one or two sidebands.

Nonlinear WGM resonators with the third order nonlinearities, such asCaF2 WGM resonators, can be used to construct tunable optical combgenerators. A CaF2 WGM resonator was used to generate optical combs with25 m GHz frequency spacing (m is an integer number). The spacing (thenumber m) was changed controllably by selecting the proper detuning ofthe carrier frequency of the pump laser with respect to a selected WGMfrequency. Demodulation of the optical comb by means of a fastphotodiode can be used to generate high-frequency microwave signals atthe comb repetition frequency or the comb spacing. The linewidth ofgenerated 25 GHz signals was less than 40 Hz.

Such a comb generator includes a laser to produce the pump laser beam, anonlinear WGM resonator and an optical coupling module to couple thepump laser beam into the nonlinear WGM resonator and to couple light outof the nonlinear WGM resonator. Tuning of the frequencies in the opticalcomb can be achieved by tuning the frequency of the pump laser beam andthe comb spacing can be adjusted by locking the pump laser to thenonlinear WGM resonator and controlling the locking condition of thepump laser.

FIG. 9 shows an example of such a comb generator. Pump light from thelaser, e.g., a 1550 nm tunable laser coupled to a fiber, is sent into aCaF2 WGM resonator using a coupling prism, and was retrieved out of theresonator using another coupling prism. The light escaping the prism iscollimated and coupled into a single mode fiber. The coupling efficiencycan be set, e.g., higher than 35%. The resonator may have a conicalshape with the rounded and polished rim. The CaF2 WGM resonator used inour tests is 2.55 mm in diameter and 0.5 mm in thickness. The intrinsicQ factor was on the order of 2.5×10⁹. The proper shaping of theresonator can reduce the mode cross section area to less than a hundredof square microns. The resonator can be packaged into a thermallystabilized box, e.g., by using a thermoelectric cooler (TEC), tocompensate for external thermal fluctuations. The optical output of theresonator was directed to an optical spectrum analyzer (OSA) to measurethe optical spectral properties of the output and a photodetector and anRF spectrum analyzer (RFSA) to measure the RF or microwave spectralproperties of the output of the photodetector.

In FIG. 9, the laser frequency is locked to a mode of the WGM resonator.As illustrated, a Pound-Drever-Hall laser feedback locking system isused where a part of the optical output of the WGM resonator is used asthe optical feedback for laser locking. The level and the phase of thelaser locking are set to be different for the oscillating andnonoscillating resonator. Increasing the power of the locked laser abovethe threshold of the oscillation causes the lock instability. This isexpected since the symmetry of the resonance changes at the oscillationthreshold. The lock parameters can be modified or adjusted whileincreasing the laser power to keep the laser locked. While the laser islocked to the WGM resonator, the detuning of the laser frequency fromthe resonance frequency of the WGM resonator can be changed to tune thecomb by modifying the lock parameters.

When the WGM resonator is optically pumped at a low input level when thepumping power approaches the threshold of the hyperparametricoscillations, no optical comb is generated and a competition ofstimulated Raman scattering (SRS) and the FWM processes is observed. TheWGM resonator used in our tests had multiple mode families of high QWGMs. We found that SRS has a lower threshold compared with the FWMoscillation process in the case of direct pumping of the modes thatbelong to the basic mode sequence. This is an unexpected result becausethe SRS process has a somewhat smaller threshold compared with thehyperparametric oscillation in the modes having identical parameters.The discrepancy is due to the fact that different mode families havedifferent quality factors given by the field distribution in the mode,and positions of the couplers. The test setup was arranged in such a waythat the basic sequence of the WGMs had lower Q factor (higher loading)compared with the higher order transverse modes. The SRS process startsin the higher-Q modes even though the modes have larger volume V. Thishappens because the SRS threshold power is inversely proportional toVQ².

Pumping of the basic mode sequence with larger power of light typicallyleads to hyperparametric oscillation taking place along with the SRS.FIG. 10 shows a measured frequency spectrum of the SRS at about 9.67 THzfrom the optical carrier and hyperparametric oscillations observed inthe CaF2 resonator pumped to a mode belonging to the basic modesequence. The structure of the lines is shown by inserts below thespectrum. The loaded quality factor Q was 10⁹ and the pump power sent tothe modes was 8 mW. Our tests indicated that hyperparametric and SRSprocesses start in the higher Q modes. The frequency separation betweenthe modes participating in these processes is much less than the FSR ofthe resonator and the modes are apparently of transverse nature. Thisalso explains the absence of FWM between the SRS light and the carrier.

The photon pairs generated by FWM are approximately 8 THz apart from thepump frequency as shown in FIG. 10. This is because the CaF2 has itszero-dispersion point in the vicinity of 1550 nm. This generation ofphoton pairs far away from the pump makes the WGM resonator-basedhyperparametric oscillator well suited for quantum communication andquantum cryptography networks. The oscillator avoids large couplinglosses occurring when the photon pairs are launched into communicationfibers, in contrast with the traditional twin-photon sources based onthe χ(2)down-conversion process. Moreover, a lossless separation of thenarrow band photons with their carrier frequencies several terahertzapart can be readily obtained.

In the tests conducted, optical combs were generated when the pump powerincreased far above the oscillation threshold. Stable optical combs weregenerated when the frequency of the laser was locked to a high Qtransverse WGM. In this way, we observed hyperparametric oscillationwith a lower threshold compared with the SRS process. Even a significantincrease of the optical pump power did not lead to the onset of the SRSprocess because of the fast growth of the optical comb lines.

FIG. 11 shows examples of hyperparametric oscillation observed in theresonator pumped with 10 mW of 1550 nm light. Spectra (a) and (b)correspond to different detuning of the pump from the WGM resonantfrequency. The measured spectrum (a) shows the result of the photonsummation process when the carrier and the first Stokes sideband,separated by 25 GHz, generate photons at 12.5 GHz frequency. The processis possible because of the high density of the WGMs and is forbidden inthe single mode family resonators.

The growth of the combs has several peculiarities. In some cases, asignificant asymmetry was present in the growth of the signal and idlersidebands as shown in FIG. 11. This asymmetry is not explained with theusual theory of hyper-parametric oscillation which predicts generationof symmetric sidebands. One possible explanation for this is the highmodal density of the resonator. In the experiment the laser pumps not asingle mode, but a nearly degenerate mode cluster. The transverse modefamilies have slightly different geometrical dispersion so the shape ofthe cluster changes with frequency and each mode family results in itsown hyperparametric oscillation. The signal and idler modes of thoseoscillations are nearly degenerate so they can interfere, andinterference results in sideband suppression on either side of thecarrier. This results in the “single sideband” oscillations that wereobserved in our tests. The interfering combs should not be considered asindependent because the generated sidebands have a distinct phasedependence, as is shown in generation of microwave signals by combdemodulation.

FIG. 12 shows (a) the optical comb generated by the CaF2 WGM resonatorpumped at by a pump laser beam of 50 mW in power, and (b) the enlargedcentral part of the measurement in (a). The generated optical comb hastwo definite repetition frequencies equal to one and four FSRs of theresonator. FIG. 13 shows the modification of the comb shown in FIG. 12when the level and the phase of the laser lock were changed. FIG. 13( b)shows the enlarged central part of the measurement in FIG. 13( a).

The interaction of the signal and the idler harmonics becomes morepronounced when the pump power is further increased beyond the pumpthreshold at which the single sideband oscillation is generated. FIGS.12 and 13 show observed combs with more than 30 THz frequency span. Theenvelopes of the combs are modulated and the reason for the modulationcan be deduced from FIG. 13( b). The comb is generated over a modecluster that changes its shape with frequency.

The above described nonlinear WGM resonator-based optical comb generatorcan be tuned and the controllable tuning of the comb repetitionfrequency is achieved by changing the frequency of the pump laser.Keeping other experimental conditions unchanged (e.g., the temperatureand optical coupling of the resonator), the level and the phase of thelaser lock can be changed to cause a change in the comb frequencyspacing. The measurements shown in FIGS. 11-13 provide examples for thetuning. This tuning capability of nonlinear WGM resonator-based combgenerators is useful in various applications.

Another feature of nonlinear WGM resonator-based comb generators is thatthe different modes of the optical comb are coherent. The demodulationof the Kerr (hyperparametric) frequency comb so generated can bedirectly detected by a fast photodiode to produce a high frequency RF ormicrowave signal at the comb repetition frequency. This is a consequenceand an indication that the comb lines are coherent. The spectral purityof the signal increases with increasing Q factor of the WGMs, theoptical power of the generated sidebands, and the spectral width of thecomb. The output of the fast photodiode is an RF or microwave beatsignal caused by coherent interference between different spectralcomponents in the comb. To demonstrate the coherent properties of thecomb, a comb with the primary frequency spacing of 25 GHz was directedinto a fast 40-GHz photodiode with an optical band of 1480-1640 nm. FIG.14 shows the recorded the microwave beat signal output by the 40-GHzphotodiode. FIG. 14( a) shows the signal in the logarithmic scale andFIG. 14( b) shows the same signal in the linear scale. FIG. 14( c) showsthe spectrum of the optical comb directed into the 40-GHz photodiode.The result of the linear fit of the microwave line indicates that thegenerated microwave beat signal has a linewidth less than 40 Hz,indicating high coherence of the beat signal. A microwave spectrumanalyzer (Agilent 8564A) used in this experiment has a 10 Hz videobandwidth, no averaging, and the internal microwave attenuation is 10 dB(the actual microwave noise floor is an order of magnitude lower). Nooptical postfiltering of the optical signal was involved.

FIG. 14 also indicates that the microwave signal is inhomogeneouslybroadened to 40 Hz. The noise floor corresponds to the measurementbandwidth (approximately 4 Hz). The broadening comes from thethermorefractive jitter of the WGM resonance frequency with respect tothe pump laser carrier frequency. The laser locking circuit based on8-kHz modulation used in the test is not fast enough to compensate forthis jitter. A faster lock (e.g., 10 MHz) may be used to allow measuringa narrower bandwidth of the microwave signal.

The comb used in the microwave generation in FIG. 14( c) has anasymmetric shape. Unlike the nearly symmetric combs in FIGS. 12 and 13,this comb is shifted to the blue side of the carrier. To produce thecomb in FIG. 14( c), the laser was locked to one of the modes belongingto the basic mode sequence. We observed the two mode oscillation processas in FIG. 10 for lower pump power that transformed into the equidistantcomb as the pump power was increased. The SRS process was suppressed.

In a different test, an externally modulated light signal was sent tothe nonlinear WGM resonator as the optical pump. FIG. 15 shows measuredchaotic oscillations measured in the optical output of the nonlinear WGMresonator. The resonator was pumped with laser light at 1550 nm that ismodulated at 25 786 kHz and has a power of 50 mW. The generated spectrumis not noticeably broader than the spectrum produced with a cw pumpedresonator and the modes are not equidistant.

Therefore, optical frequency combs can be generated by optically pumpinga WGM crystalline resonator to provide tunable comb frequency spacingcorresponding to the FSR of the resonator. The combs have large spectralwidths(e.g., exceeding 30 THz) and good relative coherence of the modes.The properties of the generated combs depend on the selection of theoptically pumped mode, and the level and the phase of the lock of thelaser to the resonator.

The above described generation of optical combs using optical cubicnonlinearity in WGM resonators can use laser locking to stabilize thefrequencies of the generated optical comb signals. As illustrated inFIG. 9, a Pound-Drever-Hall (PDH) laser feedback locking scheme can beused to lock the laser that produces the pump light to the nonlinear WGMresonator. The PDH locking is an example of laser locking techniquesbased on a feedback locking circuit that uses the light coupled of theresonator to produce an electrical control signal to lock the laser tothe resonator. The level and the phase of the lock are different for theoscillating and non-oscillating resonators. Increasing the power of thelocked laser above the threshold of the oscillation causes the lockinstability. This locking of the laser can facilitate generation ofspectrally pure microwave signals. Tests indicate that the unlocked combsignals tend to have border linewidths (e.g., about MHz) than linewidthsgenerated by a comb generator with a locked laser, e.g., less than 40 Hzas shown in FIG. 14.

Alternative to the Pound-Drever-Hall (PDH) laser feedback locking,Rayleigh scattering inside a WGM resonator or a solid state ringresonator can be used to lock a laser to such a resonator in a form ofself injection locking. This injection locking locks a laser to anonlinear resonator producing a hyperparametric frequency comb byinjecting light of the optical output of the nonlinear resonator underoptical pumping by the laser light of the laser back into the laserunder a proper phase matching condition. The optical phase of thefeedback light from the nonlinear resonator to the laser is adjusted tomeet the phase matching condition.

Two feedback mechanisms can be used to direct light from the nonlinearresonator to the laser for locking the laser. The first feedbackmechanism uses the signal produced via Rayleigh scattering inside thenonlinear resonator. The light caused by the Rayleigh scattering tracesthe optical path of the original pump light from the laser to travelfrom the nonlinear resonator to the laser.

The second feedback mechanism uses a reflector, e.g., an additionalpartially transparent mirror, placed at the output optical path of thenonlinear resonator to generate a reflection back to the nonlinearresonator and then to the laser. FIG. 16 shows an example of a device1600 that locks a laser 1601 to a nonlinear resonator 1610. Thenonlinear resonator 1610 can be a ring resonator, a disk resonator, aspherical resonator or non-spherical resonator (e.g., a spheroidresonator). An optical coupler 1620, which can be a coupling prism asshown, is used to provide optical input to the resonator 1610 and toprovide optical output from the resonator 1610. The laser 1601 producesand directs a laser beam 1661 to the coupling prism 1620 which couplesthe laser beam 1661 into the resonator 1610 as the beam 1662 circulatingin the counter-clock wise direction inside the resonator 1610. The lightof the circulating beam 1662 is optically coupled out by the opticalcoupler 1620 as a resonator output beam 1663. A reflector 1640 is placedafter the coupling prism 1620 in the optical path of the resonatoroutput beam 1663 to reflect at least a portion of the resonator outputbeam 1663 back to the coupling prism 1620. Optical collimators 1602 and1631 can be used to collimate the light. The reflector 1640 can be apartial reflector to transmit part of the resonator output beam 1663 asan output beam 1664 and to reflect part of the resonator output beam asa returned beam 1665. The reflector 1640 may also be a full reflectorthat reflects all light of the beam 1663 back as the returned beam 1665.The feedback beam 1665 is coupled into the resonator 1610 as a counterpropagating beam 1666 which is coupled by the coupling prism 1620 as afeedback beam 1667 towards the laser 1601. The feedback beam 1667 entersthe laser 1601 and causes the laser to lock to the resonator 1610 viainjecting locking.

The above laser locking based on optical feedback from the nonlinearresonator 1610 based on either the Rayleigh scattering inside theresonator 1610 or the external reflector 1640 can be established whenthe optical phase of the feedback beam 1667 from the resonator 1610 tothe laser 1601 meets the phase matching condition for the injectionlocking. A phase control mechanism can be implemented in the opticalpath of the feedback beam 1667 in the Rayleigh scattering scheme or oneor more beams 1661, 1662, 1663, 1665, 1666 and 1667 in the scheme usingthe external reflector 1640 to adjust and control the optical phase ofthe feedback beam 1667. As illustrated, in one implementation of thisphase control mechanism, the reflector 1540 may be a movable mirror thatcan be controlled to change its position along the optical path of thebeam 1663 to adjust the optical phase of the feedback beam 1667. Thephase of the returning signal 1667 can also be adjusted either by aphase rotator 1603 placed between the laser 1601 and the coupler 1620 ora phase rotator 1663 placed between the coupler 1620 or collimator 1631and the external reflector or mirror 1640. A joint configuration ofusing both the Rayleigh scattering inside the resonator 1610 and theexternal reflector 1640 may also be used. The selection of theconfiguration depends on the operating conditions including the loadingof the resonator 1610 with the coupler 1620 as well as the strength ofthe Rayleigh scattering in the resonator 1610. Such a locking techniquecan be used allow avoiding technical difficulties associated with usingthe PDH locking and other locking designs.

While this document contains many specifics, these should not beconstrued as limitations on the scope of an invention or of what may beclaimed, but rather as descriptions of features specific to particularembodiments of the invention. Certain features that are described inthis document in the context of separate embodiments can also beimplemented in combination in a single embodiment. Conversely, variousfeatures that are described in the context of a single embodiment canalso be implemented in multiple embodiments separately or in anysuitable subcombination. Moreover, although features may be describedabove as acting in certain combinations and even initially claimed assuch, one or more features from a claimed combination can in some casesbe excised from the combination, and the claimed combination may bedirected to a subcombination or a variation of a subcombination.

Only a few implementations are disclosed. Variations and enhancements ofthe described implementations and other implementations can be madebased on what is described and illustrated in this document.

1. A method for generating an RF or microwave oscillation, comprising:coupling a laser beam from a laser into a whispering-gallery-moderesonator formed of a nonlinear optical material with a third ordernonlinearity at a power level above a pump threshold power level tocause an optical hyperparametric oscillation based on a nonlinear mixingin the resonator; coupling light out of the resonator into aphotodetector to produce an RF or microwave signal at an RF or microwavefrequency with low noise; and locking the laser to the resonator tostabilize the RF or microwave signal output by the photodetector,wherein: the laser is locked to the resonator via injection locking bydirecting a portion of the light coupled out of the resonator back intothe laser.
 2. The method as in claim 1, comprising: directly couplingthe laser to the resonator to couple the laser beam into the resonator;and directly coupling the photodetector to the resonator to receive thelight coupled out of the resonator.
 3. A method for generating an RF ormicrowave oscillation, comprising: coupling a laser beam from a laserinto a whispering-gallery-mode resonator formed of a nonlinear opticalmaterial with a third order nonlinearity at a power level above a pumpthreshold power level to cause an optical hyperparametric oscillationbased on a nonlinear mixing in the resonator; coupling light out of theresonator into a photodetector to produce an RF or microwave signal atan RF or microwave frequency with low noise; locking the laser to theresonator to stabilize the RF or microwave signal output by thephotodetector; and coupling light generated by Rayleigh scatteringinside the resonator out of the resonator as a feedback beam into thelaser to lock the laser by injection locking.
 4. A method for generatingan RF or microwave oscillation, comprising: coupling a laser beam from alaser into a whispering-gallery-mode resonator formed of a nonlinearoptical material with a third order nonlinearity at a power level abovea pump threshold power level to cause an optical hyperparametricoscillation based on a nonlinear mixing in the resonator; coupling lightout of the resonator into a photodetector to produce an RF or microwavesignal at an RF or microwave frequency with low noise; locking the laserto the resonator to stabilize the RF or microwave signal output by thephotodetector; and reflecting at least a portion of the light coupledout of the resonator back to the resonator; coupling the reflectedportion into the resonator in a whispering gallery mode that propagatesin a direction opposite to a propagation direction of light that iscoupled into the resonator from the laser beam; and coupling light ofthe reflected portion coupled into the resonator out of the resonator asa feedback beam along an optical path of the laser beam from the laserin an opposite direction of the laser beam to inject the light into thelaser for locking the laser.
 5. The method as in claim 4, comprising:controlling a phase of the feedback beam to establish a phase matchingfor locking the laser to the resonator.
 6. The method as in claim 1,comprising: tuning a frequency of the laser beam generated by the laserto tune the frequency of the RF or microwave signal out of thephotodetector.
 7. The method as in claim 1, comprising: using the lightcoupled out of the resonator to generate an optical comb of differentoptical frequencies.
 8. A device for generating an RF or microwaveoscillation, comprising: a whispering-gallery-mode resonator formed of anonlinear optical material exhibiting a third order nonlinearity; alaser to produce a laser beam that interacts with the nonlinear opticalmaterial to cause an optical hyperparametric oscillation due to anonlinear mixing in the resonator; a first optical coupler that directlycouples to the laser at one end of the first optical coupler to receivethe laser beam and directly couples to the resonator to couple the laserbeam into the resonator; a second optical coupler that directly couplesto the resonator to output light out of the resonator; a photodetectordirectly coupled to the second optical coupler to receive the outputlight and to produce an optical hyperparametric oscillation signal at anRF or microwave frequency with low noise; and a laser locking mechanismthat locks the laser to the resonator, wherein: the laser lockingmechanism is based on injection locking by injecting light coupled outof the resonator into the laser.
 9. A device, comprising: an opticalresonator formed of a nonlinear optical material exhibiting a thirdorder nonlinearity and structured to circulate light in one or moreresonator modes; a laser to produce a laser beam that interacts with thenonlinear optical material to cause an optical hyperparametricoscillation due to a nonlinear mixing in the resonator; an opticalcoupler that couples to the resonator to couple the laser beam from thelaser along a first optical path into the resonator and couples lightinside the resonator out as a resonator output beam along a secondoptical path; an optical reflector in the second optical path to reflectat least a portion of the resonator output beam back to the opticalcoupler which couples the reflected portion into the resonator in a waythat the optical coupler couples light of the reflected portion insidethe resonator out as a feedback beam to the laser; and a mechanism tocontrol a phase of the feedback beam to establish a phase matching forlocking the laser to the resonator based on injection locking.
 10. Thedevice as in claim 9, comprising: a phase rotator in at least one of thefirst optical path between the laser and the resonator and the secondoptical path between the laser and the reflector to control a phase ofthe feedback beam for establishing the phase matching.
 11. The device asin claim 9, wherein: the reflector is movable in position along thesecond optical path.
 12. The device as in claim 9, wherein: theresonator is a ring resonator.
 13. The device as in claim 9, wherein:the resonator is a whispering gallery mode resonator.
 14. The method asin claim 3, comprising: directly coupling the laser to the resonator tocouple the laser beam into the resonator; and directly coupling thephotodetector to the resonator to receive the light coupled out of theresonator.
 15. The method as in claim 4, comprising: directly couplingthe laser to the resonator to couple the laser beam into the resonator;and directly coupling the photodetector to the resonator to receive thelight coupled out of the resonator.
 16. The method as in claim 3,comprising: tuning a frequency of the laser beam generated by the laserto tune the frequency of the RF or microwave signal out of thephotodetector.
 17. The method as in claim 3, comprising: using the lightcoupled out of the resonator to generate an optical comb of differentoptical frequencies.
 18. The method as in claim 4, comprising: tuning afrequency of the laser beam generated by the laser to tune the frequencyof the RF or microwave signal out of the photodetector.
 19. The methodas in claim 4, comprising: using the light coupled out of the resonatorto generate an optical comb of different optical frequencies.